The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the k > 2 largest eigenvalues of the normalized adjacency matrix (equivalently, the k smallest eigenvalues of the normalized graph Laplacian) are computed. It is shown that the information necessary for partitioning is contained in the subspace spanned by the k eigenvectors. The partitioning is encoded in a matrix $\Psi$ in indicator form, which is computed by approximating the eigenvector matrix by a product of $\Psi$ and an orthogonal matrix. A measure of the distance of a graph to being k-partitionable is defined, as well as two cut (cost) functions, for which Cheeger inequalities are proved; thus the relation between the eigenvalue and partitioning problems is established. Numerical examples are given that demonstrate that the partitioning algorithm is efficient and robust.

翻译：本文研究无向图的多路分割问题。采用谱方法，计算归一化邻接矩阵（等价于归一化图拉普拉斯矩阵）中最大的k>2个特征值（即最小的k个特征值）。结果表明，分割所需信息包含在由k个特征向量张成的子空间中。该分割通过一个指示形式的矩阵$\Psi$进行编码，该矩阵通过将特征向量矩阵近似为$\Psi$与一个正交矩阵的乘积得到。定义了图可k分割性的距离度量以及两个割（代价）函数，并证明了其Cheeger不等式；由此建立了特征值与分割问题之间的关系。数值实例表明，该分割算法高效且鲁棒。